1. **State the problem:** Gerald wants to transform the quadratic parent function $f(x) = x^2$ by reflecting it across the x-axis, stretching it vertically by a factor of 2, shifting it right by 3 units, and shifting it down by 4 units. 2. **Recall the transformation rules for quadratic functions:** - Reflection across the x-axis: multiply the function by $-1$. - Vertical stretch by a factor of $a$: multiply the function by $a$. - Horizontal shift right by $h$: replace $x$ with $(x - h)$. - Vertical shift down by $k$: subtract $k$ from the function. 3. **Apply each transformation step-by-step:** - Start with $f(x) = x^2$. - Reflect across x-axis: $-x^2$. - Stretch vertically by 2: $-2x^2$. - Shift right by 3: replace $x$ with $(x - 3)$, so $-2(x - 3)^2$. - Shift down by 4: subtract 4, so $-2(x - 3)^2 - 4$. 4. **Final transformed function:** $$g(x) = -2(x - 3)^2 - 4$$ 5. **Match with given options:** Option A matches the transformed function. **Answer:** A